Exact density of sphere packings in undiscovered dimensions

Determine the exact maximal packing density θ(d) of congruent spheres in every dimension d for which the value is not yet known.

Background

The sphere-packing problem asks for the densest possible packing of congruent spheres in Euclidean space. Exact optimal densities are known in dimensions 1, 2, 3, 8, and 24, but the paper states that no exact value is known in the other dimensions. Determining these densities is presented as a central unresolved problem in discrete geometry and has connections to coding theory.

References

The dimensions $d=8$ and $d=24$ required another breakthough due to Viazovska,, but $\theta(d)$ is not known exactly for any other dimension.

— The hard-core model in graph theory  (2501.03379 - Davies et al., 6 Jan 2025) in Section “Sphere packing”